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Annali di Matematica Pura ed Applicata

, Volume 109, Issue 1, pp 1–22 | Cite as

Non-linear operators on sets of measures

  • Richard A. Alò
  • Charles A. Cheney
  • André de Korvin
Article

Summary

If M[, U(C, C)] is the collection of U(C, C)-valued (non-linear) set functions defined on the Borel subsets of the compact Hausdorff space S, one may define operators on M[, U(C, C)] which are « of the Hammerstein type ». We initiate a study of a concept analogous to the second dual of a space of continuous functions by inquiring as to what representation theorems one may obtain for these operators. A « Lebesgue type » decomposition theorem for elements of M[, U(C, C)] is obtained. A « density » theorem is also obtained for the space M[, U(C, C)].

Keywords

Continuous Function Decomposition Theorem Hausdorff Space Compact Hausdorff Space Hammerstein Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bibliography

  1. [1]
    R. A. Alò -A. de Korvin,Approximate integration, J. Math. Analysis and its Applications48 (1974), pp. 127–138.CrossRefGoogle Scholar
  2. [2]
    J. Batt,Non-linear integral operators on C(S, E), Studia Matematica, (to appear).Google Scholar
  3. [3]
    J. Batt,Strongly additive transformations and integral representations with measures of non-linears operators, Bull. Amer. Math. Soc.,78 (1972).Google Scholar
  4. [4]
    N. Dinculeanu,Vector Measures, Pergamon Press, Berlin, 1967.Google Scholar
  5. [5]
    N. Dunford -J. Schwartz,Linear Operators, Part I, Interscience Publishers, New York, 1958.Google Scholar
  6. [6]
    A. Grothendieck,Sur les applications lineaires faiblement compactes d'espaces du type C(K), Can. J. Math.,5 (1953), pp. 129–173.zbMATHMathSciNetGoogle Scholar
  7. [7]
    S. Kaplan,On the second dual of the space of continuous functions, Trans. Amer. Math. Soc.,86 (1957), pp. 70–90.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    M. A. Krasnosel'skii,Topological methods in the theory of non-linear integral equations, GITTL, Moscow, 1956. (English translation, Macmillan, New York, 1964).Google Scholar
  9. [9]
    M. Leove,Probability Theory, D. Van Nostrand, Princeton, 1963.Google Scholar
  10. [10]
    D. Mauldin,A representation for the second dual of C[0, 1], Studia Matematica, (to appear).Google Scholar
  11. [11]
    V. J. Mizel,Characterization of non-linear transformations possessing kernels, Can. J. Math.,22 (1970), pp. 449–471.zbMATHMathSciNetGoogle Scholar
  12. [12]
    V. J. Mizel -K. Sundaresan,Representation of vector valued non-linear functions, Trans. Amer. Math. Soc.,159 (1971), pp. 111–127.CrossRefMathSciNetGoogle Scholar
  13. [13]
    M. A. Rieffel,The Radon-Nikodym theorem for the Bochner integral, Trans. Amer. Math. Soc.,131 (1968), pp. 466–487.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    S. Wayment,Absolute continuity and the Radon theorem, Ph.D. Thesis, University of Utah, 1968.Google Scholar

Copyright information

© Fondazione Annali di Matematica Para ed Applicata 1975

Authors and Affiliations

  • Richard A. Alò
    • 1
  • Charles A. Cheney
    • 1
  • André de Korvin
    • 2
  1. 1.PittsburghU.S.A.
  2. 2.Terre HauteU.S.A.

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